Equivalence of Definitions of Constant Polynomial

Theorem

Let $R$ be a commutative ring with unity.

Let $P\in R[x]$ be a polynomial in one variable over $R$.

The following definitions of the concept of Constant Polynomial are equivalent:

Definition 1

The polynomial $P$ is a constant polynomial if and only if its coefficients of $x^k$ are zero for $k \ge 1$.

Definition 2

The polynomial $P$ is a constant polynomial if and only if $P$ is either the zero polynomial or has degree $0$.

Definition 3

The polynomial $P$ is a constant polynomial if and only if it is in the image of the canonical embedding $R \to R \sqbrk x$.

Proof

1 iff 2

This is by definition of coefficients and degree.

$\Box$

1 iff 3

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